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<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.internals.series_evaluation"></a><a class="link" href="series_evaluation.html" title="Series Evaluation">Series Evaluation</a>
</h3></div></div></div>
<h5>
<a name="math_toolkit.internals.series_evaluation.h0"></a>
        <span class="phrase"><a name="math_toolkit.internals.series_evaluation.synopsis"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.synopsis">Synopsis</a>
      </h5>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">tools</span><span class="special">/</span><span class="identifier">series</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span>
</pre>
<pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">tools</span><span class="special">{</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">V</span><span class="special">&amp;</span> <span class="identifier">init_value</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">V</span><span class="special">&gt;</span>
<span class="keyword">inline</span> <span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">U</span><span class="special">&amp;</span> <span class="identifier">tolerance</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>

<span class="comment">//</span>
<span class="comment">// The following interfaces are now deprecated:</span>
<span class="comment">//   </span>
<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">U</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">,</span> <span class="identifier">U</span> <span class="identifier">init_value</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">Functor</span><span class="special">&gt;</span>
<span class="keyword">typename</span> <span class="identifier">Functor</span><span class="special">::</span><span class="identifier">result_type</span> <span class="identifier">kahan_sum_series</span><span class="special">(</span><span class="identifier">Functor</span><span class="special">&amp;</span> <span class="identifier">func</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&amp;</span> <span class="identifier">max_terms</span><span class="special">);</span>

<span class="special">}}}</span> <span class="comment">// namespaces</span>
</pre>
<h5>
<a name="math_toolkit.internals.series_evaluation.h1"></a>
        <span class="phrase"><a name="math_toolkit.internals.series_evaluation.description"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.description">Description</a>
      </h5>
<p>
        These algorithms are intended for the <a href="http://en.wikipedia.org/wiki/Series_%28mathematics%29" target="_top">summation
        of infinite series</a>.
      </p>
<p>
        Each of the algorithms takes a nullary-function object as the first argument:
        the function object will be repeatedly invoked to pull successive terms from
        the series being summed.
      </p>
<p>
        The second argument is the precision required, summation will stop when the
        next term is less than <span class="emphasis"><em>tolerance</em></span> times the result. The
        deprecated versions of <code class="computeroutput"><span class="identifier">sum_series</span></code>
        take an integer number of bits here - internally they just convert this to
        a tolerance and forward the call.
      </p>
<p>
        The third argument <span class="emphasis"><em>max_terms</em></span> sets an upper limit on
        the number of terms of the series to evaluate. In addition, on exit the function
        will set <span class="emphasis"><em>max_terms</em></span> to the actual number of terms of
        the series that were evaluated: this is particularly useful for profiling
        the convergence properties of a new series.
      </p>
<p>
        The final optional argument <span class="emphasis"><em>init_value</em></span> is the initial
        value of the sum to which the terms of the series should be added. This is
        useful in two situations:
      </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
            Where the first value of the series has a different formula to successive
            terms. In this case the first value in the series can be passed as the
            last argument and the logic of the function object can then be simplified
            to return subsequent terms.
          </li>
<li class="listitem">
            Where the series is being added (or subtracted) from some other value:
            termination of the series will likely occur much more rapidly if that
            other value is passed as the last argument. For example, there are several
            functions that can be expressed as <span class="emphasis"><em>1 - S(z)</em></span> where
            S(z) is an infinite series. In this case, pass -1 as the last argument
            and then negate the result of the summation to get the result of <span class="emphasis"><em>1
            - S(z)</em></span>.
          </li>
</ul></div>
<p>
        The two <span class="emphasis"><em>kahan_sum_series</em></span> variants of these algorithms
        maintain a carry term that corrects for roundoff error during summation.
        They are inspired by the <a href="http://en.wikipedia.org/wiki/Kahan_Summation_Algorithm" target="_top"><span class="emphasis"><em>Kahan
        Summation Formula</em></span></a> that appears in <a href="http://docs.sun.com/source/806-3568/ncg_goldberg.html" target="_top">What
        Every Computer Scientist Should Know About Floating-Point Arithmetic</a>.
        However, it should be pointed out that there are very few series that require
        summation in this way.
      </p>
<h5>
<a name="math_toolkit.internals.series_evaluation.h2"></a>
        <span class="phrase"><a name="math_toolkit.internals.series_evaluation.examples"></a></span><a class="link" href="series_evaluation.html#math_toolkit.internals.series_evaluation.examples">Examples</a>
      </h5>
<p>
        These examples are all in <a href="../../../../example/series.cpp" target="_top">../../example/series.cpp</a>
      </p>
<p>
        Let's suppose we want to implement <span class="emphasis"><em>log(1+x)</em></span> via its
        infinite series,
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="inlinemediaobject"><img src="../../../equations/log1pseries.svg"></span>

        </p></blockquote></div>
<p>
        We begin by writing a small function object to return successive terms of
        the series:
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">log1p_series</span>
<span class="special">{</span>
   <span class="comment">// we must define a result_type typedef:</span>
   <span class="keyword">typedef</span> <span class="identifier">T</span> <span class="identifier">result_type</span><span class="special">;</span>

   <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
      <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span> <span class="special">{}</span>

   <span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()()</span>
   <span class="special">{</span>
      <span class="comment">// This is the function operator invoked by the summation</span>
      <span class="comment">// algorithm, the first call to this operator should return</span>
      <span class="comment">// the first term of the series, the second call the second</span>
      <span class="comment">// term and so on.</span>
      <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
      <span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="special">++</span><span class="identifier">k</span><span class="special">;</span>
   <span class="special">}</span>

<span class="keyword">private</span><span class="special">:</span>
   <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
   <span class="keyword">const</span> <span class="identifier">T</span> <span class="identifier">m_mult</span><span class="special">;</span>
   <span class="identifier">T</span> <span class="identifier">m_prod</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
        Implementing log(1+x) is now fairly trivial:
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">T</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">x</span><span class="special">)</span>
<span class="special">{</span>
   <span class="comment">// We really should add some error checking on x here!</span>
   <span class="identifier">BOOST_MATH_ASSERT</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>

   <span class="comment">// Construct the series functor:</span>
   <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
   <span class="comment">// Set a limit on how many iterations we permit:</span>
   <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">max_iter</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
   <span class="comment">// Add it up, with enough precision for full machine precision:</span>
   <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">(),</span> <span class="identifier">max_iter</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<p>
        We can almost use the code above for complex numbers as well - unfortunately
        we need a slightly different definition for epsilon, and within the functor,
        mixed complex and integer arithmetic is sadly not supported (as of C++17),
        so we need to cast out integers to floats:
      </p>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="keyword">struct</span> <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span>
<span class="special">{</span>
   <span class="comment">// we must define a result_type typedef:</span>
   <span class="keyword">typedef</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">result_type</span><span class="special">;</span>

   <span class="identifier">log1p_series</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
      <span class="special">:</span> <span class="identifier">k</span><span class="special">(</span><span class="number">0</span><span class="special">),</span> <span class="identifier">m_mult</span><span class="special">(-</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">m_prod</span><span class="special">(-</span><span class="number">1</span><span class="special">)</span> <span class="special">{}</span>

   <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="keyword">operator</span><span class="special">()()</span>
   <span class="special">{</span>
      <span class="comment">// This is the function operator invoked by the summation</span>
      <span class="comment">// algorithm, the first call to this operator should return</span>
      <span class="comment">// the first term of the series, the second call the second</span>
      <span class="comment">// term and so on.</span>
      <span class="identifier">m_prod</span> <span class="special">*=</span> <span class="identifier">m_mult</span><span class="special">;</span>
      <span class="keyword">return</span> <span class="identifier">m_prod</span> <span class="special">/</span> <span class="identifier">T</span><span class="special">(++</span><span class="identifier">k</span><span class="special">);</span>
   <span class="special">}</span>

<span class="keyword">private</span><span class="special">:</span>
   <span class="keyword">int</span> <span class="identifier">k</span><span class="special">;</span>
   <span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_mult</span><span class="special">;</span>
   <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">m_prod</span><span class="special">;</span>
<span class="special">};</span>


<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">&gt;</span>
<span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">log1p</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="identifier">x</span><span class="special">)</span>
<span class="special">{</span>
   <span class="comment">// We really should add some error checking on x here!</span>
   <span class="identifier">BOOST_MATH_ASSERT</span><span class="special">(</span><span class="identifier">abs</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">&lt;</span> <span class="number">1</span><span class="special">);</span>

   <span class="comment">// Construct the series functor:</span>
   <span class="identifier">log1p_series</span><span class="special">&lt;</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;</span> <span class="special">&gt;</span> <span class="identifier">s</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span>
   <span class="comment">// Set a limit on how many iterations we permit:</span>
   <span class="identifier">std</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">max_iter</span> <span class="special">=</span> <span class="number">1000</span><span class="special">;</span>
   <span class="comment">// Add it up, with enough precision for full machine precision:</span>
   <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">sum_series</span><span class="special">(</span><span class="identifier">s</span><span class="special">,</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special">&lt;</span><span class="identifier">T</span><span class="special">&gt;::</span><span class="identifier">epsilon</span><span class="special">()),</span> <span class="identifier">max_iter</span><span class="special">);</span>
<span class="special">}</span>
</pre>
<p>
        Of course with a few traits classes and a bit of meta-programming we could
        fold these two implementations into one, but that's beyond the scope of these
        examples.
      </p>
</div>
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
<td align="left"></td>
<td align="right"><div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div></td>
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